Arithmetic Functions and Primes

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Harmonic Analysis is to be enhanced with combinatorial ideas and applied to develop representations by products and quotients evaluated at the primes. Immediate aims include the complete multiplicative generation of the positive rationals by shifted primes, and the solution of a thirty year old problem of Katai concerning sums of additive functions on arithmetic progressions. An ultimate aim is the celebrated problem of Goldbach and the infinitude of twin primes. The value distribution of primitive roots is also to be studied. The project aims to develop in number theory a flexible general theory of arithmetic functions strong enough to attack certain classically difficult problems. All three of the topics to be considered directly or indirectly involve the multiplicative generation of rationals by given rationals of a restricted form. Positive results have immediate relevance to the construction of algorithms to factorise integers and to the theoretical study of encryption based upon the difficulties of factorising large numbers.